Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games

Codenotti, Luca; Lewicka, Marta; Manfredi, Juan

doi:10.1090/tran/6962

Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games
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by Luca Codenotti, Marta Lewicka and Juan Manfredi PDF

Trans. Amer. Math. Soc. 369 (2017), 7387-7403 Request permission

Abstract:

We study the double-obstacle problem for the $p$-Laplace operator, $p\in [2, \infty )$. We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-of-war games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token’s position, converges to $0$. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.

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